Impact of spatial clustering on disease transmission and optimal control

Abstract

Spatial heterogeneities and spatial separation of hosts are often seen as key factors when developing accurate predictive models of the spread of pathogens. The question we address in this paper is how coarse the resolution of the spatial data can be for a model to be a useful tool for informing control policies. We examine this problem using the specific case of foot-and-mouth disease spreading between farms using the formulation developed during the 2001 epidemic in the United Kingdom. We show that, if our model is carefully parameterized to match epidemic behavior, then using aggregate county-scale data from the United States is sufficient to closely determine optimal control measures (specifically ring culling). This result also holds when the approach is extended to theoretical distributions of farms where the spatial clustering can be manipulated to extremes. We have therefore shown that, although spatial structure can be critically important in allowing us to predict the emergent population-scale behavior from a knowledge of the individual-level dynamics, for this specific applied question, such structure is mostly subsumed in the parameterization allowing us to make policy predictions in the absence of high-quality spatial information. We believe that this approach will be of considerable benefit across a range of disciplines where data are only available at intermediate spatial scales.

Fig. 1.

Graphs showing the average density of farms against radius around each farm as the radius varies for the true data (blue line) and random data (red line) in (A) Cumbria, UK and (B) Lancaster County, PA. Insets show farm locations for each respective county for the true data (left plot) and the random data (right plot). The color scale on the insets shows the number of cattle on each farm. (C) Average density of farms against radius around each farm for the true data for Devon, Aberdeenshire, and Clwyd in the United Kingdom and Cuming, Wright, Humboldt, and Franklin in the United States.

Fig. 2.

Epidemic impact against ring-cull radius for epidemics in (A) Cumbria and (B) Lancaster County. In both figures, the blue line shows the mean epidemic impact for simulations using the true location data while the red line shows the mean epidemic impact for the reparameterized random data. Both lines are calculated as locally smooth splines fit to 10,000 simulation results. (C) Farm network and (D) mean epidemic impact against ring-cull radius for the random data (red line) and the true data (blue line). For (C) and (D), S₀ = 4, N = 1,000, and B = 0.4. In (A), (B), and (D), the black dots show the minima of each line.

Fig. 3.

Using the full clustered data, these graphs show the impact of ring culling at the true (R_T) and approximated (R_RR) optimal radius, as the distribution of farms controlled by the parameter B and the ratio S₀∶S_inf vary. In (A) and (B), the random-location model is fitted to the entire epidemic derived from a simulation using the spatially clustered location data, while in (C) and (D), only the first 14 days of the epidemic are used to fit the random-location model. In (A) and (C), the color scale gives the percentage of farms that would be saved by additional ring culling at radius R_RR compared with IP and DC culling alone. In (B) and (D), the color scale gives the additional saving in the epidemic impact when ring culling at the true optimal radius (R_T) is compared with at R_RR.